Basic use
Compute a lazy Kronecker products between two matrices A and B by either
K = kronecker(A, B)or, by using the binary operator:
K = A ⊗ BNote, ⊗ can be formed by typing \otimes<tab>.
The Kronecker product K behaves like a matrix, for which size(K), eltype(K) works as one would expect. Elements can be accessed via K[i,j]; every element is computed on the fly. The function collect can be used to turn K in a regular, dense matrix.
julia> using Kronecker
julia> A = randn(4, 4)
4×4 Array{Float64,2}:
1.57575 -0.501709 -3.21159 1.47518
-0.226798 1.29403 -0.864597 -0.645007
1.11364 0.142855 -0.345008 2.77911
1.63773 1.08353 -0.140607 1.93866
julia> B = rand(1:10, 5, 7)
5×7 Array{Int64,2}:
2 3 10 7 6 10 2
2 9 3 10 10 10 7
7 3 5 8 5 4 10
4 6 7 2 10 2 10
8 9 5 8 3 8 6
julia> K = A ⊗ B
20×28 Kronecker.KroneckerProduct{Float64,Array{Float64,2},Array{Int64,2}}:
3.15151 4.72726 15.7575 11.0303 … 8.8511 14.7518 2.95037
3.15151 14.1818 4.72726 15.7575 14.7518 14.7518 10.3263
11.0303 4.72726 7.87876 12.606 7.37592 5.90074 14.7518
6.30301 9.45452 11.0303 3.15151 14.7518 2.95037 14.7518
12.606 14.1818 7.87876 12.606 4.42555 11.8015 8.8511
-0.453596 -0.680395 -2.26798 -1.58759 … -3.87004 -6.45007 -1.29001
-0.453596 -2.04118 -0.680395 -2.26798 -6.45007 -6.45007 -4.51505
-1.58759 -0.680395 -1.13399 -1.81439 -3.22504 -2.58003 -6.45007
-0.907193 -1.36079 -1.58759 -0.453596 -6.45007 -1.29001 -6.45007
-1.81439 -2.04118 -1.13399 -1.81439 -1.93502 -5.16006 -3.87004
2.22728 3.34092 11.1364 7.79548 … 16.6747 27.7911 5.55822
2.22728 10.0228 3.34092 11.1364 27.7911 27.7911 19.4538
7.79548 3.34092 5.5682 8.90912 13.8956 11.1164 27.7911
4.45456 6.68184 7.79548 2.22728 27.7911 5.55822 27.7911
8.90912 10.0228 5.5682 8.90912 8.33733 22.2329 16.6747
3.27546 4.91319 16.3773 11.4641 … 11.632 19.3866 3.87732
3.27546 14.7396 4.91319 16.3773 19.3866 19.3866 13.5706
11.4641 4.91319 8.18866 13.1018 9.6933 7.75464 19.3866
6.55092 9.82639 11.4641 3.27546 19.3866 3.87732 19.3866
13.1018 14.7396 8.18866 13.1018 5.81598 15.5093 11.632
julia> K[4, 5]
15.7575287024678
julia> eltype(K) # promotion
Float64
julia> collect(K)
20×28 Array{Float64,2}:
3.15151 4.72726 15.7575 11.0303 … 8.8511 14.7518 2.95037
3.15151 14.1818 4.72726 15.7575 14.7518 14.7518 10.3263
11.0303 4.72726 7.87876 12.606 7.37592 5.90074 14.7518
6.30301 9.45452 11.0303 3.15151 14.7518 2.95037 14.7518
12.606 14.1818 7.87876 12.606 4.42555 11.8015 8.8511
-0.453596 -0.680395 -2.26798 -1.58759 … -3.87004 -6.45007 -1.29001
-0.453596 -2.04118 -0.680395 -2.26798 -6.45007 -6.45007 -4.51505
-1.58759 -0.680395 -1.13399 -1.81439 -3.22504 -2.58003 -6.45007
-0.907193 -1.36079 -1.58759 -0.453596 -6.45007 -1.29001 -6.45007
-1.81439 -2.04118 -1.13399 -1.81439 -1.93502 -5.16006 -3.87004
2.22728 3.34092 11.1364 7.79548 … 16.6747 27.7911 5.55822
2.22728 10.0228 3.34092 11.1364 27.7911 27.7911 19.4538
7.79548 3.34092 5.5682 8.90912 13.8956 11.1164 27.7911
4.45456 6.68184 7.79548 2.22728 27.7911 5.55822 27.7911
8.90912 10.0228 5.5682 8.90912 8.33733 22.2329 16.6747
3.27546 4.91319 16.3773 11.4641 … 11.632 19.3866 3.87732
3.27546 14.7396 4.91319 16.3773 19.3866 19.3866 13.5706
11.4641 4.91319 8.18866 13.1018 9.6933 7.75464 19.3866
6.55092 9.82639 11.4641 3.27546 19.3866 3.87732 19.3866
13.1018 14.7396 8.18866 13.1018 5.81598 15.5093 11.632Constructing Kronecker products
Kronecker.kronecker — Functionkronecker(A::AbstractMatrix, B::AbstractMatrix)Construct a Kronecker product object between two arrays. Does not evaluate the Kronecker product explictly.
kronecker(A::AbstractMatrix, B::AbstractMatrix)Higher-order Kronecker lazy kronecker product, e.g.
kronecker(A, B, C, D)kronecker(A::AbstractMatrix, pow::Int)Kronecker power, computes A ⊗ A ⊗ ... ⊗ A. Returns a lazy KroneckerPower type.
Kronecker.:⊗ — Function⊗(A::AbstractMatrix, B::AbstractMatrix)Binary operator for kronecker, computes as Lazy Kronecker product. See kronecker for documentation.
⊗(A::AbstractMatrix, pow::Int)Kronecker power, computes A ⊗ A ⊗ ... ⊗ A. Returns a lazy KroneckerPower type.
Base.collect — Methodcollect(K::GeneralizedKroneckerProduct)Collects a lazy instance of the GeneralizedKroneckerProduct type into a dense, native matrix. Falls back to the element-wise case when not specialized method is defined.
Basic properties of Kronecker products
Base.getindex — Functiongetindex(K::AbstractKroneckerProduct, i1::Integer, i2::Integer)Computes and returns the (i,j)-th element of an AbstractKroneckerProduct K. Uses recursion if K is of an order greater than two.
Missing docstring for eltype. Check Documenter's build log for details.
Base.size — Methodsize(K::AbstractKroneckerProduct)Returns a the size of an AbstractKroneckerProduct instance.
Base.collect — Functioncollect(K::GeneralizedKroneckerProduct)Collects a lazy instance of the GeneralizedKroneckerProduct type into a dense, native matrix. Falls back to the element-wise case when not specialized method is defined.
collect(K::AbstractKroneckerSum)Collects a lazy instance of the AbstractKroneckerSum type into a full, native matrix. Returns the result as a sparse matrix.
function collect(E::Eigen{<:Number, <:Number, <:AbstractKroneckerProduct})Collects eigenvalue decomposition of a AbstractKroneckerProduct type into a matrix.
Kronecker.collect! — Functioncollect!(C::AbstractMatrix, K::GeneralizedKroneckerProduct)In-place collection of K in C. If possible, specialized routines are used to speed up the computation. The fallback is an element-wise iteration. In this case, this function might be slow.
collect!(C::AbstractMatrix, K::AbstractKroneckerProduct)In-place collection of K in C where K is an AbstractKroneckerProduct, i.e., K = A ⊗ B.
collect!(C::AbstractMatrix, K::AbstractKroneckerSum)In-place collection of K in C where K is an AbstractKroneckerSum, i.e., K = A ⊗ B.
Kronecker.order — Functionorder(M::AbstractMatrix)Returns the order of a matrix, i.e. how many matrices are involved in the Kronecker product (default to 1 for general matrices).
Kronecker.getmatrices — Functiongetmatrices(K::AbstractKroneckerProduct)Obtain the two matrices of an AbstractKroneckerProduct object.
getmatrices(A::AbstractArray)Returns a matrix itself. Needed for recursion.
getmatrices(K::T) where T <: AbstractKroneckerSumObtain the two matrices of an AbstractKroneckerSum object.
Kronecker.issquare — Functionissquare(A::AbstractMatrix)Checks if an array is a square matrix.
issquare(A::Factorization)Checks if a Factorization struct represents a square matrix.
LinearAlgebra.issymmetric — Functionissymmetric(K::AbstractKroneckerProduct)Checks if a Kronecker product is symmetric.
Missing docstring for sum. Check Documenter's build log for details.
Linear algebra
Many functions of the LinearAlgebra module are overloaded to work with subtypes of GeneralizedKroneckerProduct.
LinearAlgebra.det — Methoddet(K::AbstractKroneckerProduct)Compute the determinant of a Kronecker product.
LinearAlgebra.logdet — Methodlogdet(K::AbstractKroneckerProduct)Compute the logarithm of the determinant of a Kronecker product.
LinearAlgebra.tr — Methodtr(K::AbstractKroneckerProduct)Compute the trace of a Kronecker product.
Base.inv — Methodinv(K::AbstractKroneckerProduct)Compute the inverse of a Kronecker product.
Base.adjoint — Methodadjoint(K::AbstractKroneckerProduct)Compute the adjoint of a Kronecker product.
Base.transpose — Methodtranspose(K::AbstractKroneckerProduct)Compute the transpose of a Kronecker product.
Missing docstring for conj(K::AbstractKroneckerProduct). Check Documenter's build log for details.